Answer :
To find the quotient of the polynomial division, let's start by reviewing the given problem:
We have two polynomials:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
We need to divide the dividend by the divisor to find the quotient. The division will result in a quotient and possibly a remainder, but we're focused on finding the quotient here.
Here’s a step-by-step explanation of how polynomial long division works in this case:
1. Divide the leading term of the dividend by the leading term of the divisor:
Divide [tex]\(x^4\)[/tex] (the leading term of the dividend) by [tex]\(x^3\)[/tex] (the leading term of the divisor). This gives us [tex]\(x\)[/tex].
2. Multiply the entire divisor by this result:
Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
3. Subtract this result from the original dividend:
Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]. This subtraction leaves you with:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
Which simplifies to:
[tex]\[
5x^3 - 15
\][/tex]
4. Repeat the division process for the new polynomial:
Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives us [tex]\(5\)[/tex].
5. Multiply the divisor by this new result:
Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], which results in [tex]\(5x^3 - 15\)[/tex].
6. Subtract this from the current polynomial:
Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since there is nothing left over, it shows that the remainder is 0.
The quotient from this division is:
- [tex]\(x + 5\)[/tex]
This process confirms that our solution to the division is [tex]\(x + 5\)[/tex], and since there's no remainder, the quotient is indeed a complete representation of the division of these polynomials.
We have two polynomials:
- Dividend: [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]
- Divisor: [tex]\(x^3 - 3\)[/tex]
We need to divide the dividend by the divisor to find the quotient. The division will result in a quotient and possibly a remainder, but we're focused on finding the quotient here.
Here’s a step-by-step explanation of how polynomial long division works in this case:
1. Divide the leading term of the dividend by the leading term of the divisor:
Divide [tex]\(x^4\)[/tex] (the leading term of the dividend) by [tex]\(x^3\)[/tex] (the leading term of the divisor). This gives us [tex]\(x\)[/tex].
2. Multiply the entire divisor by this result:
Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(x\)[/tex], resulting in [tex]\(x^4 - 3x\)[/tex].
3. Subtract this result from the original dividend:
Subtract [tex]\(x^4 - 3x\)[/tex] from [tex]\(x^4 + 5x^3 - 3x - 15\)[/tex]. This subtraction leaves you with:
[tex]\[
(x^4 + 5x^3 - 3x - 15) - (x^4 - 3x) = 5x^3 + 0x^2 + 0x - 15
\][/tex]
Which simplifies to:
[tex]\[
5x^3 - 15
\][/tex]
4. Repeat the division process for the new polynomial:
Divide the new leading term [tex]\(5x^3\)[/tex] by [tex]\(x^3\)[/tex], which gives us [tex]\(5\)[/tex].
5. Multiply the divisor by this new result:
Multiply [tex]\(x^3 - 3\)[/tex] by [tex]\(5\)[/tex], which results in [tex]\(5x^3 - 15\)[/tex].
6. Subtract this from the current polynomial:
Subtract [tex]\(5x^3 - 15\)[/tex] from [tex]\(5x^3 - 15\)[/tex]:
[tex]\[
(5x^3 - 15) - (5x^3 - 15) = 0
\][/tex]
Since there is nothing left over, it shows that the remainder is 0.
The quotient from this division is:
- [tex]\(x + 5\)[/tex]
This process confirms that our solution to the division is [tex]\(x + 5\)[/tex], and since there's no remainder, the quotient is indeed a complete representation of the division of these polynomials.
